This isn't the Rust experience. Rust still distributes clippy as completely separate from cargo, and rustfmt as separate too, etc. it's technically only rustup that centralizes the distribution of the tools together.
Nevertheless, since rustup is most folks entry point to the rust ecosystem, they are effectively distributed together, and rustfmt is fully integrated in cargo.
cargo-fmt is not part of Cargo, it's distributed with rustfmt and uses the same mechanism as third-party subcommands like cargo-expand. Neither are cargo-clippy and cargo-miri.
(Cargo does, however, contain a small list of those official subcommands so that if you try to use one that's missing, it can remind you to install the corresponding component.)